分野別セミナー

We consider the supercritical bond percolation on the Z^d lattice and study the chemical distance, i.e., the graph distance on the infinite cluster. It is well-known from the Kingman subadditive ergodic theorem that there exists a deterministic constant μ(x) such that the chemical distance D(0, nx) between two connected points 0 and nx grows like nμ(x). Garet and Marchand prove that the probability of the upper tail large deviation event {D(0, nx) > μ(nx)(1 + ε), 0 ↔ nx} decays exponentially with n. In this talk, we discuss the existence of the rate function for the upper tail large deviation when d ≥ 3 and ε > 0 is small enough. Moreover, for d ≥ 3, we prove that the upper tail large deviation event is created by a space-time cut-point (a point that any geodesic from 0 to nx must cross at a given time) that forces the geodesics to go in a non-optimal direction or to wiggle before reaching the cut-point, where the geodesics lose extra time. This enables us to express the rate function in terms of the rate function for a space-time cut-point. This talk is based on joint work with Barbara Dembin (ETH).